Nabarro–Herring Creep
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materials science Materials science is an interdisciplinary field of researching and discovering materials. Materials engineering is an engineering field of finding uses for materials in other fields and industries. The intellectual origins of materials sci ...
, Nabarro–Herring creep (NH creep) is a
mechanism Mechanism may refer to: *Mechanism (economics), a set of rules for a game designed to achieve a certain outcome **Mechanism design, the study of such mechanisms *Mechanism (engineering), rigid bodies connected by joints in order to accomplish a ...
of deformation of
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
line materials (and
amorphous In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid) is a solid that lacks the long-range order that is a characteristic of a crystal. The terms "glass" and "glassy solid" are sometimes used synonymousl ...
materials) that occurs at low stresses and held at elevated temperatures in
fine-grained Granularity (also called graininess) is the degree to which a material or system is composed of distinguishable pieces, "granules" or "grains" (metaphorically). It can either refer to the extent to which a larger entity is subdivided, or the ...
materials. In Nabarro–Herring creep, atoms
diffuse Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
through the crystals, and the rate of creep varies inversely with the square of the
grain size Grain size (or particle size) is the diameter of individual grains of sediment, or the lithified particles in clastic rocks. The term may also be applied to other granular materials. This is different from the crystallite size, which ...
so fine-grained materials creep faster than coarser-grained ones. NH creep is solely controlled by diffusional mass transport. This type of creep results from the diffusion of vacancies from regions of high
chemical potential In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
at
grain boundaries In materials science, a grain boundary is the interface between two grains, or crystallites, in a polycrystalline material. Grain boundaries are two-dimensional crystallographic defect, defects in the crystal structure, and tend to decrease the ...
subjected to normal tensile stresses to regions of lower chemical potential where the average tensile stresses across the grain boundaries are zero. Self-diffusion within the grains of a
polycrystalline A crystallite is a small or even microscopic crystal which forms, for example, during the cooling of many materials. Crystallites are also referred to as grains. Bacillite is a type of crystallite. It is rodlike with parallel longulites. S ...
solid can cause the solid to yield to an applied
shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
, the yielding being caused by a diffusional flow of matter within each crystal grain away from boundaries where there is a normal pressure and toward those where there is a normal tension. Atoms migrating in the opposite direction account for the creep strain (). The creep strain rate is derived in the next section. NH creep is more important in
ceramics A ceramic is any of the various hard, brittle, heat-resistant, and corrosion-resistant materials made by shaping and then firing an inorganic, nonmetallic material, such as clay, at a high temperature. Common examples are earthenware, porce ...
than
metals A metal () is a material that, when polished or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. These properties are all associated with having electrons available at the Fermi level, as against no ...
as dislocation motion is more difficult to effect in ceramics.


Derivation of the creep rate

Source: The Nabarro–Herring creep rate, \dot\varepsilon_ , can be derived by considering an individual rectangular grain (in a single or polycrystal). Two opposing sides have a
compressive stress Compressive stresses are generated in objects when they are subjected to forces that push inward, causing the material to shorten or compress. These stresses occur when an object is squeezed or pressed from opposite directions. In everyday life, ...
applied and the other two have a
tensile stress In continuum mechanics, stress is a physical quantity that describes forces present during deformation. For example, an object being pulled apart, such as a stretched elastic band, is subject to ''tensile'' stress and may undergo elongati ...
applied. The atomic volume is decreased by compression and increased by tension. Under this change, the
activation energy In the Arrhenius model of reaction rates, activation energy is the minimum amount of energy that must be available to reactants for a chemical reaction to occur. The activation energy (''E''a) of a reaction is measured in kilojoules per mole (k ...
to form a vacancy is altered by \pm \sigma\Omega . The atomic volume is \Omega and the stress is \sigma . The plus and minus indication is an increase or decrease in the activation energy due to the tensile and compressive stresses, respectively. The fraction of vacancy concentrations in the compressive ( N_^C ) and tensile ( N_^T ) regions are given as: :\begin N_^C &\approx \exp \left(-\frac \right) \exp \left(-\frac \right) \\ pt N_^T &\approx \exp \left(-\frac \right) \exp \left(\frac \right) \end In these equations Q_f is the vacancy formation energy, k is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
, and T is the absolute
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
. These vacancy concentrations are maintained at the lateral and horizontal surfaces in the grain. These net concentrations drive vacancies to the compressive regions from the tensile ones which causes grain elongation in one dimension and grain compression in the other. This is creep deformation caused by a flux of vacancy motion. The vacancy
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phe ...
, J_ , associated with this motion is given by: : J_ = -D_ \left(\frac\right) where D_ is the vacancy diffusivity. This is given as: : D_ = D_ \exp \left(\frac \right) where D_ is the diffusivity when 0 vacancies are present and Q_m is the vacancy motion energy. The term \tfrac is the vacancy concentration gradient. The term \delta x is proportional to the grain size d and \delta N_= N_^T - N_^C . If we multiply J_ by d^2 we obtain: : \frac \approx D_ d \exp \left(- \frac \right) \left exp\left(\frac \right)- \exp\left( -\frac\right) \right where \tfrac is the volume changed per unit time during creep deformation. The change in volume can be related to the change in length along the tensile axis as \delta V \approx d^2 \delta d . Using the relationship between \delta V and \delta d the NH creep rate is given by: :\begin \dot\varepsilon_ &= \frac\frac \\ pt \dot\varepsilon_ &= \frac \exp \left(-\frac \right) \left exp \left(\frac\right) -\exp\left(-\frac \right)\right \end This equation can be greatly simplified. The lattice self-diffusion coefficient is given by: : D_L = D_ \exp \left(-\frac \right) As previously stated, NH creep occurs at low stresses and high temperatures. In this range \sigma \Omega << kT . For small x , \exp(\pm x) \approx 1 \pm x . Thus we can re-write \dot\varepsilon_ as: : \dot\varepsilon_ = A_ \left(\frac\right)\left(\frac \right) where A_ is a constant that absorbs the approximations in the derivation. Alternatively, this can be derived in a different method where the constant A_n has different dimensions. In this case, the NH creep rate \dot\varepsilon is given by: :\dot\varepsilon = \frac


Comparison to Coble creep

Coble creep In materials science, Coble creep, a form of diffusion creep, is a mechanism for deformation of crystalline solids. Contrasted with other diffusional creep mechanisms, Coble creep is similar to Nabarro–Herring creep in that it is dominant at ...
is closely related to Nabarro–Herring creep and is controlled by diffusion as well. Unlike Nabarro–Herring creep, mass transport occurs by diffusion along the surface of single crystals or the grain boundaries in a polycrystal. For a general expression of creep rate, the comparison between Nabarro–Herring and Coble creep can be presented as follows: :\dot\varepsilon = \frac^n^p is the
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the Elasticity (physics), elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear s ...
. The diffusivity is obtained form the tracer diffusivity, D^*. The dimensionless constant A_n depends intensively on the geometry of grains. The parameters A, n and p are dependent on creep mechanisms. Nabbaro–Herring creep does not involve the motion of dislocations. It predominates over high-temperature dislocation-dependent mechanisms only at low stresses, and then only for fine-grained materials. Nabarro–Herring creep is characterized by creep rates that increase linearly with the stress and inversely with the square of grain diameter. In contrast, in
Coble creep In materials science, Coble creep, a form of diffusion creep, is a mechanism for deformation of crystalline solids. Contrasted with other diffusional creep mechanisms, Coble creep is similar to Nabarro–Herring creep in that it is dominant at ...
atoms diffuse along grain boundaries and the creep rate varies inversely with the cube of the grain size. Lower temperatures favor Coble creep and higher temperatures favor Nabbaro–Herring creep because the activation energy for vacancy diffusion within the lattice is typically larger than that along the grain boundaries, thus lattice diffusion slows down relative to grain boundary diffusion with decreasing temperature.


Experimental and theoretical examples

* Creep in dense, polycrystalline magnesium oxide and iron-doped polycrystalline magnesia * Compressive creep in polycrystalline beryllium oxide * Creep in polycrystalline that has been doped with Cr, Fe, or Ti * Creep in dry synthetic dunite which results in trace melt and some grain growth * Reproduced for nanopolycrystalline systems in Phase Field Crystal simulations (theory matched in terms of creep stress and grain size exponents)


References

* * * {{DEFAULTSORT:Herring-Nabarro creep Materials degradation